Recently, constructions of eigenvalue processes which satisfy SDEs (REF ) for general \(\beta >0\)  are reported. As mentioned above, it is clear for \(\beta =1,2\)  and 4. Allez and Guionnet [1]} constructed a sequence of symmetric (hermitian) matrix-valued processes whose eigenvalue processes converge weakly to a stochastic process, and the limiting process satisfies (REF ) for \(0<\beta \le 1\)  (\(0<\beta \le 2\) , respectively) with the Ornstein-Uhlenbeck drift. Holcomb and Paquette consider a symmetric tridiagonal matrix-valued process whose eigenvalue process satisfies (REF ) (or (REF )) and give a sufficient condition of such matrices in [2]}. Their construction relies on the Lanczos algorithm. Fukushima, Tanida and Yano proved [3]} that a \(2\times 2\)  symmetric matrix-valued process, which is defined by putting independent Brownian motions and Bessel processes on diagonal entries and off-diagonal ones, respectively, has the eigenvalue process satisfying (REF ) with general \(\beta >0\) . Note that this symmetric matrix-valued process is a natural time dependent model of Gaussian beta ensemble (G\(\beta \) E for short), introduced by Dumitriu and Edelman [4]}. G\(\beta \) E is a well-known model because this is a generalization of GOE, GUE and GSE in the following sense.
For \(\beta >0\) , G\(\beta \) E is an \(N \times N \)  symmetric tridiagonal matrix \(H_{\beta }^{tri}\)  defined by
\(H_{\beta }^{tri}=\dfrac{1}{\sqrt{\beta }}\begin{pmatrix}g_1 & \chi _{(N-1)\beta } & 0 & \cdots & \cdots & 0\\\chi _{(N-1)\beta } & g_2 & \chi _{(N-2)\beta } & & & \vdots \\0 & \chi _{(N-2)\beta } & & \ddots & & \vdots \\\vdots & & \ddots & \ddots & \chi _{2\beta }& 0\\\vdots & & & \chi _{2\beta } & g_{N-1} & \chi _{\beta }\\0 & \cdots &\cdots & 0 & \chi _{\beta } & g_N\end{pmatrix},\) 
